We follow the notation and terminology of [1]}. A linear forest is a forest whose connected components are paths. A star forest is a forest whose connected components are stars. A path-star forest is a forest whose connected components are paths and stars. In 2011, Bushaw and Kettle [2]} determined the Turán numbers of \(kP_{\ell }\)  for sufficiently large \(n\) , which was extended by Lidiciký, Liu and Palmer [1]}. Yuan and Zhang [4]}, [5]} determined the Turán numbers of linear forests containing at most one odd path for all \(n\) . The following lemma is based on Theorem 1.7 of [5]}.
